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Theorem nbgrael 21430
 Description: The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.)
Assertion
Ref Expression
nbgrael Neighbors

Proof of Theorem nbgrael
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 21425 . . . 4 Neighbors
21mpt2xopn0yelv 6456 . . 3 Neighbors
32pm4.71rd 617 . 2 Neighbors Neighbors
4 nbgraop 21428 . . . . . 6 Neighbors
54eleq2d 2502 . . . . 5 Neighbors
6 preq2 3876 . . . . . . 7
76eleq1d 2501 . . . . . 6
87elrab 3084 . . . . 5
95, 8syl6bb 253 . . . 4 Neighbors
109pm5.32da 623 . . 3 Neighbors
11 3anass 940 . . 3
1210, 11syl6bbr 255 . 2 Neighbors
133, 12bitrd 245 1 Neighbors
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  crab 2701  cpr 3807  cop 3809   crn 4871  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340   Neighbors cnbgra 21422 This theorem is referenced by:  nbgrasym  21441  nbgraf1olem1  21443  usg2spot2nb  28391 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-nbgra 21425
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